Harmonic measures versus quasiconformal measures for hyperbolic groups

Blachère, Sébastien; Haïssinsky, Peter; Mathieu, Pierre

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 4, pp. 683-721.

On établit une formule de la dimension de la mesure harmonique d'une marche aléatoire de loi de support fini et symétrique sur un groupe hyperbolique. On caractérise aussi les lois pour lesquelles la dimension est maximale. Notre approche repose sur la distance de Green, une distance qui permet de développer un point de vue géométrique sur les marches aléatoires et, en particulier, d'interpréter les mesures harmoniques comme des mesures quasiconformes.

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.

Classification : 20F67, 60B15, 11K55, 20F69, 28A75, 60J50, 60J65
Keywords: hyperbolic groups, random walks on groups, harmonic measures, quasiconformal measures, dimension of a measure, Martin boundary, brownian motion, Green metric
Mot clés : groupes hyperboliques, marches aléatoires sur les groupes, mesures harmoniques, mesures quasiconformes, dimension d'une mesure, bord de Martin, mouvement brownien, distance de Green

[1] A. Ancona, Positive harmonic functions and hyperbolicity, in Potential theory-surveys and problems (Prague, 1987), Lecture Notes in Math. 1344, Springer, 1988, 1-23. | MR | Zbl

[2] A. Ancona, Théorie du potentiel sur les graphes et les variétés, in École d'été de Probabilités de Saint-Flour XVIII-1988, Lecture Notes in Math. 1427, Springer, 1990, 1-112. | MR | Zbl

[3] M. T. Anderson & R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. 121 (1985), 429-461. | MR | Zbl

[4] W. Ballmann, On the Dirichlet problem at infinity for manifolds of nonpositive curvature, Forum Math. 1 (1989), 201-213. | MR | Zbl

[5] W. Ballmann & F. Ledrappier, Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, in Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sémin. Congr. 1, Soc. Math. France, 1996, 77-92. | MR | Zbl

[6] G. Besson, G. Courtois & S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995), 731-799. | MR | Zbl

[7] M. Björklund, Central limit theorems for Gromov hyperbolic groups, J. Theoret. Probab. 23 (2010), 871-887. | MR | Zbl

[8] S. Blachère & S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probab. Theory Related Fields 137 (2007), 323-343. | MR | Zbl

[9] S. Blachère, P. Haïssinsky & P. Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, Ann. Probab. 36 (2008), 1134-1152. | MR | Zbl

[10] M. Bonk & O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266-306. | MR | Zbl

[11] M. Bourdon & H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in dynamics and geometry (Cambridge, 2000), Springer, 2002, 1-17. | MR | Zbl

[12] R. Bowen & C. Series, Markov maps associated with Fuchsian groups, Publ. Math. I.H.É.S. 50 (1979), 153-170. | Numdam | MR | Zbl

[13] C. Connell & R. Muchnik, Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces, Geom. Funct. Anal. 17 (2007), 707-769. | MR | Zbl

[14] M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), 241-270. | MR | Zbl

[15] B. Deroin, V. Kleptsyn & A. Navas, On the question of ergodicity for minimal group actions on the circle, Mosc. Math. J. 9 (2009), 263-303. | MR | Zbl

[16] E. B. Dynkin, The boundary theory of Markov processes (discrete case), Uspehi Mat. Nauk 24 (1969), 3-42. | MR | Zbl

[17] E. Ghys & P. De La Harpe (éds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser, 1990. | MR | Zbl

[18] Y. Guivarc'H, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, in Conference on Random Walks (Kleebach, 1979), Astérisque 74, Soc. Math. France, 1980, 47-98. | Numdam | MR | Zbl

[19] Y. Guivarc'H & Y. Le Jan, Sur l'enroulement du flot géodésique, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 645-648. | MR | Zbl

[20] Y. Guivarc'H & Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sci. École Norm. Sup. 26 (1993), 23-50. | Numdam | MR | Zbl

[21] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, 2001. | MR | Zbl

[22] G. A. Hunt, Markoff chains and Martin boundaries, Illinois J. Math. 4 (1960), 313-340. | MR | Zbl

[23] V. A. Kaĭmanovich, Brownian motion and harmonic functions on covering manifolds. An entropic approach, Soviet Math. Dokl. 33 (1986), 812-816. | MR | Zbl

[24] V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Hyperbolic behaviour of dynamical systems, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 361-393. | Numdam | MR | Zbl

[25] V. A. Kaimanovich, Discretization of bounded harmonic functions on Riemannian manifolds and entropy, in Potential theory (Nagoya, 1990), de Gruyter, 1992, 213-223. | MR | Zbl

[26] V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. reine angew. Math. 455 (1994), 57-103. | MR | Zbl

[27] V. A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), 631-660. | MR | Zbl

[28] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math. 152 (2000), 659-692. | MR | Zbl

[29] I. Kapovich & N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296, Amer. Math. Soc., 2002, 39-93. | MR | Zbl

[30] A. Karlsson & F. Ledrappier, Propriété de Liouville et vitesse de fuite du mouvement brownien, C. R. Math. Acad. Sci. Paris 344 (2007), 685-690. | MR | Zbl

[31] V. Le Prince, Dimensional properties of the harmonic measure for a random walk on a hyperbolic group, Trans. Amer. Math. Soc. 359 (2007), 2881-2898. | MR | Zbl

[32] V. Le Prince, A relation between dimension of the harmonic measure, entropy and drift for a random walk on a hyperbolic space, Electron. Commun. Probab. 13 (2008), 45-53. | MR | Zbl

[33] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds, Bol. Soc. Brasil. Mat. 19 (1988), 115-140. | MR | Zbl

[34] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275-287. | MR | Zbl

[35] F. Ledrappier, Some asymptotic properties of random walks on free groups, in Topics in probability and Lie groups: boundary theory, CRM Proc. Lecture Notes 28, Amer. Math. Soc., 2001, 117-152. | MR | Zbl

[36] R. Lyons, Equivalence of boundary measures on covering trees of finite graphs, Ergodic Theory Dynam. Systems 14 (1994), 575-597. | MR | Zbl

[37] T. Lyons & D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), 299-323. | MR | Zbl

[38] J. Mairesse & F. Mathéus, Random walks on free products of cyclic groups, J. Lond. Math. Soc. 75 (2007), 47-66. | MR | Zbl

[39] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier, Grenoble 7 (1957), 183-281. | Numdam | MR | Zbl

[40] M. A. Pinsky, Stochastic Riemannian geometry, in Probabilistic analysis and related topics, Vol. 1, Academic Press, 1978, 199-236. | MR | Zbl

[41] J.-J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), AA1539-A1542. | MR | Zbl

[42] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187-231. | MR | Zbl

[43] A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Russian Math. Surveys 55 (2000), 667-733. | MR | Zbl

[44] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138, Cambridge Univ. Press, 2000. | MR | Zbl

[45] L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124. | MR | Zbl